Show that the circumference of the Bohr orbit for the hydrogen atom is an integral multiple of the de Broglie wavelength associated with the electron revolving around the nucleus.

1. Bohr’s model of atom:

(i) Frequency of radiation absorbed or emitted during transition: $\mathrm{v}=\frac{\mathrm{\Delta E}}{\mathrm{h}}\mathrm{=}\frac{{\mathrm{E}}_{\mathrm{2}}-{\mathrm{E}}_{\mathrm{1}}}{\mathrm{h}}$

(ii) Quantization of angular momentum: $\mathrm{mvr}=\mathrm{n}\frac{\mathrm{h}}{2\mathrm{\pi }}$

(iii) Energy of stationary states: ${\mathrm{E}}_{\mathrm{n}}\mathrm{=}-\frac{21.8{\times 10}^{-\mathrm{19}}}{{\mathrm{n}}^{\mathrm{2}}}{\mathrm{Z}}^{\mathrm{2}}\hspace{0.17em}\mathrm{J}\hspace{0.17em}{\mathrm{atom}}^{-1}$

(iv) Radii of the stationary orbits: ${\mathrm{r}}_{\mathrm{n}}=52.9\times \left(\frac{{\mathrm{n}}^{\mathrm{2}}}{\mathrm{Z}}\right)\mathrm{pm}$

(v) Energy gap between two orbits: $\mathrm{\Delta E}={\mathrm{R}}_{\mathrm{H}}\left(\frac{\mathrm{1}}{{\mathrm{n}}_{\mathrm{i}}^{\mathrm{2}}}-\frac{\mathrm{1}}{{\mathrm{n}}_{\mathrm{f}}^{\mathrm{2}}}\right)$

Where, ${\mathrm{n}}_{\mathrm{i}}$ and ${\mathrm{n}}_{\mathrm{f}}$ are initial orbit and final orbit respectively.

2. De Broglie relation:

de Broglie relation is $\mathrm{\lambda } = \mathrm{h}/\mathrm{mv} = \mathrm{h}/\mathrm{p}$ where λ is wavelength, m is mass, v is velocity and p is momentum of the material particle.

3. Heisenberg's uncertainty principle:

$\mathrm{\Delta x}.\mathrm{\Delta p}>\frac{\mathrm{h}}{4\mathrm{\pi }}\text{ or }\mathrm{m\Delta x}.\mathrm{\Delta v}\ge \frac{\mathrm{h}}{4\mathrm{\pi }}\mathrm{\hspace{1em}}\text{ or }\hspace{1em}\mathrm{\Delta x}.\mathrm{\Delta v}\ge \frac{\mathrm{h}}{4\mathrm{\pi m}}$

4. Quantum numbers:

The electron in an atom can be characterised by a set of four quantum numbers, namely principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (m) and spin quantum number (s).

5. Electronic configuration:

(i) Aufbau principle : Once the lower energy orbitals are completely filled, then the electrons enter the next higher energy orbitals.

(ii) Pauli exclusion principle : No two electrons in an atom can have the same set of values of all four quantum numbers.

(iii) Hund's rule: It states that electron pairing in the degenerate orbitals does not take place until all the available orbitals contains one electron each.

(iv) Number of subshells in ${\mathrm{n}}^{\mathrm{th}}$ shell $= \mathrm{n}$

(v) Number of orbitals in ${\mathrm{n}}^{\mathrm{th}}$ shell$={\mathrm{n}}^2$

(vi) Number of electrons in ${\mathrm{n}}^{\mathrm{th}}$ shell$=2{\mathrm{n}}^2$

(vii) Number of orbitals in subshell $= 2\mathrm{l} + 1$

(viii) Number of electrons in subshell $= 2(2\mathrm{l} + 1)$

(ix) Total number of nodes in any orbital $= \mathrm{n} - 1$

(x) Number of spherical/radial nodes in any orbital $= \mathrm{n} - \mathrm{l} – 1$ where, $\mathrm{l} =$ number of angular nodes.